MAGIC SQUARES

A "magic square" is a rectangular array of numbers, usually from 1 to n²,  so that each column, row, and both diagonals have the same sum. A magic square is an arrangement of the numbers from 1 to n² in an nxn matrix, with each number occurring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum must be n(n²⁺¹)/2.

• "Odd" Magic Squares, which means that there is an odd number of cells on each side of the Magic Square.

• "Even" Magic Squares, which means that there is an even number of cells on each side of the Magic Square. "Even" Magic Squares may be further divided into two sub-categories:

The next simplest is the 3x3 magic square

Numeric Magic Squares may be divided into two categories:

"singly even" Magic Squares, which means that the number of cells on each side of the Magic Square is evenly divisible by two, but not by four (e.g. 6 x 6 and 10 x 10 Magic Squares)

"doubly even" Magic Squares, which means that the number of cells on each side of the Magic Square is evenly divisible by both two and four (e.g. 4 x 4 and 8 x 8 Magic Squares).

  

 The simplest magic square is the 1x1 magic square whose only entry is the number 1.

You can assemble the numbers 1 to 9 in a square, so that the sum of the rows, the columns, and the diagonals is 15.

                           

ü  Put the first number in the middle column of the top row.

ü  Put the next number in the box moved one column to the right and one row up.

ü  If the number exceeds a column or a row, place it in the opposite side of that column or row.

ü  Repeat step 2 'n' times just before you reach the original starting position.

ü  Place the next number in the same column one row below the last number and continue with step 2.

4x4 magic square

    The 4x4 Dürer magic square (or, what is essentially the same, the 4x4 magic square I use) has many interesting special properties that are not shared by magic squares in general. They are so interesting that they are often pointed out when this square is presented. That is good, but can sometimes lead to misunderstandings as to which is the meat and which is the gravy.

    In the case of this 4x4 magic square:

ü  In addition to having the sum 34 (= 4(4^2+1)/2) in each row, column and main diagonal,

ü  The four corners add to 34.

ü  The four numbers in the center add to 34.

ü  The 15 and 14 in the top row and the 3 and 2 facing them in the bottom row add to 34.

ü  The 12 and 8 in the first column and the 9 and 5 facing them in the last column add to 34.

ü  The four squares in the corners add to 34.

ü  If you go clockwise around the square and choose the first squares away from the corners (15,9,2,8), they add to 34. The same holds if you go counter clockwise.